Residue theorem

The residue theorem states that a positively oriented closed contour integral of a meromorphic function equals 2πi times the sum of its residues inside the contour.

Residue: The coefficient of the −1 power in a function’s Laurent series around an isolated singularity. Each enclosed singularity contributes its residue to the contour integral.
Contour integration: The evaluation of integrals along paths in the complex plane. The theorem is a central tool for evaluating closed contour integrals.
Holomorphic function: A complex function that is complex-differentiable throughout an open domain. Away from its poles, the theorem’s meromorphic integrand is holomorphic.
Cauchy's theorem (several complex variables): A family of results extending Cauchy-type integral theorems to functions of several complex variables. The one-variable residue theorem has no direct identical contour-sum form in higher dimensions.
Laurent series: A power series expansion that permits finitely many negative powers near an isolated singularity. Its coefficient of the −1 power is the local quantity summed by the theorem.
Improper integral: An integral over an unbounded interval or with an unbounded integrand, defined by a limit. Suitable contours convert some real improper integrals into residue sums.
Contour integral: An integral of a function along an oriented curve in the complex plane. The theorem concerns integrals over closed contours.
Residue calculus: Methods for evaluating integrals and studying functions by computing residues at singularities. This broader toolkit includes the theorem and practical techniques built around it.
Meromorphic function: A complex function that is holomorphic except at isolated poles. The theorem applies to these functions when their poles avoid the contour.
Jordan's lemma: A result controlling arc integrals of certain exponential functions in the upper or lower half-plane. It helps justify closing contours when residues are used for Fourier-type integrals.

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